Trope Theory, Resemblance, and Russell's Regress Florian Boge 12.07.2012 Structure • Introduction – what are tropes? – Nominalism about universals – Bundle theory – A refinement: nucleus theory • Similarity relations – A trope theoretical measure for relative resemblance • The resemblance regress – a fundamental problem for trope theory – Is it vicious? – Similarity as internal – A cognitivist approach as a possible solution • Perfect resemblance defined What are tropes? • Definition: Tropes are the particular properties (property instances) of a given concrete entity (cf. Campbell 1990, 18). They are abstract particulars. – Relatons = polyadic tropes – Qualitons = monadic tropes (cf. Bacon 2008, 2) • An entity is called abstract (in this context) iff. it is a part of some other entity, which can only be separated in thought (cf. Rojek 2008, 361). • Particulars (indviduals) = entities which only exist in one place at one time (interval) Spacio-temproal location as an important criterion for individuality according to trope theory: “[O]ur abstract particulars are particulars because they have a local habitation, even if no name. They exist as individuals at unique place- times.” (Campbell 1990, 3) What are tropes? • Examples: – The particular shape of a given chair – Bill Clinton‟s eloquence – „This redness‟, in contrast to „redness‟ in general • Supposed to provide an alternative to realism about universals Trope theory is a form of nominalism about universals Needs to explain our use of general terms Should be able to explain every day life‟s entities such as things, their appearance, their relations etc. Nominalism about universals • Nominalism about universals = attempt to provide an explanation of general terms (i.e. terms for types, properties, relations etc.) without appeal to universals • Universals = entities that are multiply exemplified i.e. exist in more than one place at the same time • Different kinds of nominalism: – Predicate nominalism: different entities just fall under the same predicate (cf. Armstrong 1978,13) highly unsatisfactory – Class nominalism: different entities are part of the same class; the predicate refers to the whole class Faces some great difficulties, e. g. coextension problem („cordate‟ and „renate‟), aggregate problem (aggregate of soldiers = aggregate of armies) Resemblance nominalism: different entities resemble each other and constitute resemblance classes (which are the denotation of a predicate) Most promising, but also has to face the coextension problem (cf. Schurz 1995, 100) Nominalism about universals • Trope theory as a form of resemblance nominalism which ipso facto solves some of the problems of classical similarity nominalism: – No coextension problem; class of resembling F-tropes distinct (even disjunct) from the class of resembling G-tropes, even if all F- and G- tropes always occur in the same classical individuals – Makes it easier to define relative resemblance of classical individuals (resemblance w.r.t. one property instead of overall resemblance) and degrees of resemblance in virtue of perfectly resembling tropes (definition will be given later) Bundle theory • Classical individuals = concrete entities we encounter in everyday life, „things‟ • How can trope theory account for those? • One classical, intuitive approach: substrata, „bare particulars‟, that bear tropes Difficulties: • „Ontological extra‟ (dispensable) • Relation of „bearing‟ must be second order trope that needs explanation: how can tropeless substratum have the property (trope) of being a bearer? (Cf. Simons 1994, 567) • Alternative: bundle theory – Tropes (always - ?) appear in bundles – Tropes in a bundle are compresent Several tropes are compresent at a time (interval) iff. they occupy the same space at that time (interval) (cf. Russell 1948, 260) Can also be interpreted to introduce a cognitive dimension; several tropes are percieved as one complex A refinement: nucleus theory • Problem: how can bundle theory account for the distinction between essential and accidental properties (i.e. tropes)? • Solution: There is a nucleus of tropes in every bundle, where any trope of the nucleus cannot exist without any of the other tropes (cf. Simons 1994, 567). One sided dependence of accidental tropes on the existence of the nucleus • Phenomenological conception of „essential‟ and „accidental‟ (cf. Gallagher and Zahavi 2008, 27). • Alternative interpretation: – Some tropes are essentially compresent: basic tropes (tropes that constitute the basic physical entities). – The tropes of several nuclei can be accidentally compresent (constitute bigger entities together) and create supervenient tropes; introduce degrees of compresence. – Many Relatons may be defined in terms of compresence (degree of compresence, change in the degree of compresence…) Less anthropocentric Similarity relations • General properties are taken to be resemblance classes of tropes – what is resemblance? – Resemblance is a similarity relation. – Technical definition: A similarity relation is a symmetric and reflexive relation:xy(Sxy Syx) x(Sxx) (cf. Bacon 2008, 4) – Resemblance as the nominalistic pendant to qualitative identity (having the same property) – Perfect resemblance is the highest degree of resemblance; it is, arguably, also transitive: xyz(Rxy Ryz Rxz) – Perfect resemblance, thus, is an equivalence relation, which induces a partition on a given Domain D. – Relative resemblance of classical individuals can be modeled in terms of perfectly resembling basic tropes; a measure for resemblance in some respect can be defined. – Compresence can – logically – be taken to be a similarity relation (similarity w.r.t. spacio-temporal location). A trope theoretical measure for relative resemblance • Motivation: „[A]n orangeness trope might be like the color of a given tomato, which in turn was like a bunch of redness tropes. The tomato‟s color might be both an orangeness and a redness. Yet the orangeness trope might not be like any of the other redness tropes. I see no compelling reason to exclude such possibilities a priori.” (Bacon 1995, 15) Argument for the non-transitivity of resemblance – plausible? Can be resolved by introducing degrees of similarity (resemblance); take resemblance in general to be fuzzy: „[A] similarity relation, S, is a fuzzy relation which is reflexive, symmetric, and transitive. Thus, let x, y be elements of a set X and s(x, y) denote the grade of membership of the ordered pair (x, y) in S. Then S is a similarity relation in X if and only if, for all x, y, z in X, (x, x) = 1 s (reflexivity), s(x, y) = s(y, x) (symmetry), and s(x, z) y (s(x, y) s(y, z)) (transitivity), where and denote max and min, respectively.” (Zadeh 1971, 177) • We need y because is a real-valued function into the closed interval [0,1]. A trope theoretical measure for relative resemblance • If we assume that the tropes of classical individuals supervene on the basic tropes, we can define the following measure for (relative) resemblance: Let x, y be variables for the elements of the set I of classical individuals and F any relation of relative resemblance on I (e.g. color resemblance). Further, let B be the set of basic tropes and ti, tj, with i, jIN , be variables for the elements of B. Let T(x) be the set of basic tropes compresent in x, i.e. t1,t2T(x): Kt1t2, where K is the relation of compresence on B. Let N = |{<ti, tj>BF² : tiT(x), tjT(y)}| where BFB is the set of basic F-tropes (the set of basic tropes on which the relata of F – e.g. color-tropes – supervene). Let P be the relation of perfect resemblance on BF. Then: F(x, y) = |{<ti, tj>P: tiT(x)BF and tjT(y)BF}| / N – Relates the idea of degrees of resemblance to trope theory – Reduces the need for a general definition of resemblance to the need for a definition of perfect resemblance – Bears some explanatory value for the question of how degrees of resemblance come into existence The resemblance regress • Resemblance relations must either consist relatons (particular resemblances; 2nd order tropes) or the universal of resemblance must be accepted. • Problem: – If we accept a universal at the basis of our ontology, trope theory does not give us a complete form of nominalism. – However, if we take resemblances themselves to be individuals, the regress problem arises: • Let r1 be the particular resemblance between two tropes a and b and r2 the particular resemblance between two tropes c and d • Now, for the r1 and r2 both to be resemblances, there must be another, higher order trope r1,2 which makes them both resemblances (since this is our criterion for membership in a resemblance class). • Let, in general, ri,j be the resemblance between two resemblances ri and rj • For any two ri,j and rk,n, we now need a r(i,j),(k,n) for them to be resemblances – and so forth for any two higher order resemblances, ad infinitum. Is it vicious? • Some philosophers have argued, that there are „virtuous‟ regresses, e.g.: – The truth regress (resulting from the substitution of the truth conditions of a true sentence into Tarskis T-scheme) – The regress of successor numbers, resulting from the axioms of peano arithmetic (cf. Nolan 2001, 523-524) • The regress of resemblances could be such a regress (cf. Campbell 1990, 35 ff.) Campbell: more formality and less substance on every level – what does that even mean? Daly (1994, 256-257): applies to any regress. We need a better argument. Resemblance as internal • Some philosophers have argued that resemblance is an internal relation (cf. Armstrong 1989, pp. 43), i.e.: – Given the existence two resembling entities (tropes) a and b, they must resemble one another (they resemble one another in every possible world, in which they exist). Tropes necessitate their resemblance Resemblance supervenes on tropes • Question: Is resemblance ontologically reducible to the resembling entities? (cf. Armstrong 1989, 56) Problem: If ra,b is the resemblance of a and b and it is „nothing besides a and b‟ then it should hold that ra,b = a and ra,b = b. But then a = b.